We can see that \(\widetilde{H}\)  is obtained
from \(H\)  by rearranging the entries in an appropriate order.
The above observation yields
\({\rm tr}_1 H = {\rm tr}_2 \widetilde{H}\) .
Moreover, it is not hard to check that
\(\widetilde{H}\)  and \(H\)  are unitarily similar;
see, e.g., [1]} or [2]}.
Motivated by these relations,
Choi [1]} introduced recently the definition of partial determinants
corresponding to partial traces. For \(H=[H_{i,j}]_{i,j=1}^n\in \mathbb {M}_n(\mathbb {M}_k)\) ,
the partial determinants are defined as
\( \mathrm {det}_1H :=\bigl [ \det G_{l,m}\bigr ]_{l,m=1}^k, \) 
